14 LOCALLY CONVEX SPACES 71 that, if A is convex and (xt)i$t$a is any finite sequence of points of A and n n is any sequence of n real numbers i>0 such that V A, = 1, we have Y A, x( e A. 1*1 *=i (a) Let E, F be vector spaces over R and let/be an affine linear mapping of E into F. Then the image under / of any convex subset of E is convex, and the inverse image under/of any convex subset of F is convex. In particular, if g ^ 0 is a linear form on E, then the half-spaces defined by one or other of the relations g(x) > a, g(x) ^ a (where a is any real number) are convex sets. (b) In a vector space E over R, the intersection of any family of convex sets is convex. If A, B are convex subsets of E and a, jS are any real numbers, then the set aA + /?B is convex. (c) Let (Ea)ae i be a family of vector spaces over R and let E = fj Ea be their product. a For each a, let Aa be a nonempty subset of E«. Show that A — f][ Aa is convex if and a only if each Aa is convex. (d) In a topological vector space E, the closure A and the interior A of a convex set A are convex. If, moreover, A is not empty, then A is the closure of A, and A the interior of A. (e) In a topological vector space E, for a half-space {x : g(x) i> a} (where g is a non- zero linear form on E) to be either closed or to have an interior point, it is necessary and sufficient that g should be continuous; in which case the hyperplane g(x) = a is the frontier of the half-space. 12. (a) Let E be a vector space over R and p a semi norm on E. Show that the set of x e E such that p(x) < 1 (resp. p(x) ^ 1) is an absorbing symmetric convex set. Con- versely, let A be an absorbing symmetric convex set in E, and for each JteE let p(x) — inf p. Show that p is a seminorm on E and that A contains the set of p>0 jcepA points such that p(x) < 1 and is contained in the set of points such that p(x) rg 1. (b) Let E be a topological vector space over R, and p a seminorm on E. Then the set A = {x e E : p(x) <£ 1} is closed if and only if p is lower semicontinuous on E. Also A has an interior point if and only if p is continuous at the point 0, and p is then continuous throughout E. (c) Deduce from (b) that a topological vector space E is locally convex if and only if there exists a fundamental system of convex neighborhoods of 0 in E. The topology of a metrizable locally convex space can be defined by an at most denumerable family of seminorms. 13. Let E be a real vector space. If A is any subset of E, the convex hull of A is defined to be the smallest convex set containing A. If E is a topological vector space, the closed convex hull of A is defined to be the smallest closed convex set containing A; it is also the closure of the convex hull of A. (a) Show that the convex hull of A is the set of all points 2 ^«x* > where (x«) is any X finite family of points of A, and the Aa are real numbers ^0 such that £ Aa » 1. a (b) Suppose that E is a metrizable locally convex space. Show that the convex hull in E of a precojnpact set is precompact. If E is a Frechet space, the closed convex hull of a compact set is compact (cf. Section 13.4, Problem 13). 14. Let E be the real vector space of regulated functions / on 1 = [0,1], continuous on the right at all / < 1 and such that/(I) = 0. For each integer n > 0, let V« be thee extended to a non-